SYSTEMS OF SIMPLE GROUPS DERIVED FROM 

 THE ORTHOGONAL GROUP. 



(Second Paper.) 



BY LEONARD EUGENE DICKSON, PH. D. 



Instructor in Mathematics, University of California. 



i. This paper completes the investigations begun in the 

 Proceedings of the California Academy of Sciences, 3d 

 Ser., Math. -physics, Vol. I, No. 4, pp. 29-46. The final 

 result may be stated as follows: — 



The squares of the linear substitutions on m indices in the 

 Galois Field of order ft 11 , p^>2, which leave the sum of the 

 squares of the m indices invariant, generate a group which 

 is simple if m be odd (with the exception p n = 3, m = 3 ), 

 but has the factors of composition 2 and one-half its order if 

 m be even and^>^, 



2. The case p" = 8 / ± 1 remains to be treated. Let 



x 6 

 O " ' denote a particular substitution not in the group Q 12 . 

 ij 2 



We study the group Hi given by extending 1 Q b}/ all the 



products O *'&. Q 7'f . 

 7 ,J A', t 



Theorem. H x contains half 1 of the substitutions of G, the 



group of all orthogonal substitutions of determinant unity. 



In fact, every substitution of G is of the form 



hi, h 2 , . . . being substitutions of H\. This product can be 



v S x B 



put into the form h O* ' . Indeed, O .' . can be carried 



i, 2 * ,J 



to the right of every Q . . and every Q , ,f t (k and I ^ i,i). 



1 For^ 11 =17 this extension is unnecessary [see appendix]. 



2 Compare § 8. 



f47] Nov. 16, 1899. 



